Question: Suppose $\cos R = \frac{4}{9}$ in the diagram below.  What is $QS$?



[asy]

pair Q,R,S;

S = (0,0);

Q = (sqrt(65),0);

R = (sqrt(65),-4);

draw(S--Q--R--S);

draw(rightanglemark(S,Q,R,13));

label("$S$",S,NW);

label("$Q$",Q,NE);

label("$R$",R,SE);

label("$9$",(R+S)/2,SW);

[/asy]
Solution: Since $\cos R = \frac{4}{9}$ and $\cos R = \frac{QR}{RS}=\frac{QR}{9}$, we have $\frac{QR}{9} = \frac{4}{9}$, so $QR = 4$. Then, by the Pythagorean Theorem, $QS = \sqrt{RS^2 - QR^2} = \sqrt{81-16} = \boxed{\sqrt{65}}$.